Recent Submissions

  • Salinization of coastal aquifers under uncertainties

    Litvinenko, Alexander (2023-05-21) [Poster]
    We consider a class of density-driven flow problems. We are particularly interested in the problem of the salinization of coastal aquifers. We consider the Henry saltwater intrusion problem with uncertain porosity, permeability, and recharge parameters as a test case. The reason for the presence of uncertainties is the lack of knowledge, inaccurate measurements,and inability to measure parameters at each spatial or time location. This problem is nonlinear and time-dependent. The solution is the salt mass fraction, which is uncertain and changes in time. Uncertainties in porosity, permeability, recharge, and mass fraction are modeled using random fields. This work investigates the applicability of the well-known multilevel Monte Carlo (MLMC) method for such problems. The MLMC method can reduce the total computational and storage costs. Moreover, the MLMC method runs multiple scenarios on different spatial and time meshes and then estimates the mean value of the mass fraction. 'The parallelization is performed in both the physical space and stochastic space. To solve every deterministic scenario, we run the parallel multigrid solver ug4 in a black-box fashion. We use the solution obtained from the quasi-Monte Carlo method as a reference solution.
  • Pricing American options under rough Heston

    Breneis, Simon (2023-05-21) [Poster]
    The rough Heston model is a popular option pricing model in mathematical finance. However, due to the non-semimartingale and non-Markovian characteristics of its volatility process, simulations can be prohibitively expensive in practice. Building on previous works, we approximate the volatility process with an N-dimensional diffusion, yielding a Markovian approximation of the rough Heston model. Then, we introduce a weak discretization scheme to simulate paths of these Markovian approximations. Our numerical experiments show that these approximations converge at a second-order rate as the number of time steps approaches infinity. We leverage these approximations to price Bermudan options under the rough Heston model.
  • Goal Oriented Adaptive Finite Element Multilevel Monte Carlo

    Liu, Yang (2023-05-21) [Poster]
    We present our work [1], Adaptive Multilevel Monte Carlo (AMLMC). It is based upon the work [2], which developed convergence rates for an adaptive algorithm based on isoparametric d-linear quadrilateral finite element approximations and the dual weighted residual error representation in the deterministic setting. The present work can also be seen as an extension of [3]. Our AMLMC algorithm uses as a building block an auxiliary sequence of deterministic, non-uniform meshes, which are generated by the above-mentioned deterministic adaptive algorithm, and they correspond to a geometrically decreasing sequence of tolerances. Specifically, for a given realization of the diffusivity coefficient and a given accuracy level, the AMLMC constructs its approximate sample as the one using the first mesh in the hierarchy that satisfies a corresponding bias accuracy constraint.
  • Uncertainty quantification in data-driven solar physics simulations

    Hall, Eric (2023-05-21) [Poster]
    Reliable and accurate predictions of solar eruptions are vital to mitigating the consequences of severe space weather, which is included in most national risk registers. Events that lead to solar eruptions are challenging to measure directly. Solar physicists rely on simulations of mathematical models, such as nonlinear force-free field (NLFFF) models, to understand the evolution of the 3D coronal magnetic configuration for a solar active region. NLFFF models are data-driven (assimilate observed magnetograms) and are used to predict time series involved in identifying possibly eruptive regions. However, dependence on initial data, assimilation timescales, measurement errors, and aleatory all contribute to uncertainty in NLFFF predictions. We report on recent progress in quantifying uncertainty in NLFFF predictions arising from misspecification of simulation start times using information theory. This is joint work with Karen Meyer (University of Dundee).
  • Double-loop quasi-Monte Carlo estimator for nested integration

    Bartuska, Arved (2023-05-21) [Poster]
    Nested integration arises when a nonlinear function is applied to an integrand, and the result is integrated again, which is common in engineering problems, such as optimal experimental design, where typically neither integral has a closed-form expression. Using the Monte Carlo method to approximate both integrals leads to a double-loop Monte Carlo estimator, which is often prohibitively expensive, as the estimation of the outer integral has bias relative to the variance of the inner integrand. For the case where the inner integrand is only approximately given, additional bias is added to the estimation of the outer integral. Variance reduction methods, such as importance sampling, have been used successfully to make computations more affordable. Furthermore, random samples can be replaced with deterministic low-discrepancy sequences, leading to quasi-Monte Carlo techniques. Randomizing the low-discrepancy sequences simplifies the error analysis of the proposed double-loop quasi-Monte Carlo estimator. To our knowledge, no comprehensive error analysis exists yet for truly nested randomized quasi-Monte Carlo estimation (i.e., for estimators with low-discrepancy sequences for both estimations). We derive asymptotic error bounds and a method to obtain the optimal number of samples for both integral approximations. Then, we demonstrate the computational savings of this approach compared to standard nested (i.e., double-loop) Monte Carlo integration when estimating the expected information gain via an example from Bayesian optimal experimental design involving an experiment from solid mechanics.
  • Learning-Based Importance Sampling via Stochastic Optimal Control for Stochastic Reaction Networks

    Wiechert, Sophia (2023-05-21) [Poster]
    We explore efficient estimation of statistical quantities, particularly rare event probabilities, for stochastic reaction networks. Consequently, we propose an importance sampling (IS) approach to improve the Monte Carlo (MC) estimator efficiency based on an approximate tau-leap scheme. The crucial step in the IS framework is choosing an appropriate change of probability measure to achieve substantial variance reduction. This task is typically challenging and often requires insights into the underlying problem. Therefore, we propose an automated approach to obtain a highly efficient path-dependent measure change based on an original connection in the stochastic reaction network context between finding optimal IS parameters within a class of probability measures and a stochastic optimal control formulation. Optimal IS parameters are obtained by solving a variance minimization problem. First, we derive an associated dynamic programming equation. Analytically solving this backward equation is challenging, hence we propose an approximate dynamic programming formulation to find near-optimal control parameters. To mitigate the curse of dimensionality, we propose a learning-based method to approximate the value function using a neural network, where the parameters are determined via a stochastic optimization algorithm. Our analysis and numerical experiments verify that the proposed learning-based IS approach substantially reduces MC estimator variance, resulting in a lower computational complexity in the rare event regime, compared with standard tau-leap MC estimators.